Calculator and decimal point locator



April 12, 1949. A, F ECKEL 2,466,883

CALCULATOR AND DECIMAL POINT LOCATOR Filed Feb. 16, 1945 2 Sheets-Sheet 1 April 12, 1949. A. F. Ecm, 2,466,883

CALCULATOR AND DECIMAL POINT LOCATOR man 1w mmm wscm Damos A I A R11-DI Patented Apr. 12, 1949 CALCULATOR .AND DECIMAL POINT LOCATOR Arthur F. Eckel, Chicago, Ill., assignor to Myrtle Scott Eckel, Chicago, and Louise Wicker-sham Pickett, La Grange, Ill.

`Application February 16, 1945, Serial No. 578,157

(Cl. 23E-64.3)

15 claims. l

This invention relates to a calculator of the slide rule Variety and a decimal point locator associated therewith whereby the decimal point in a calculation can be determined in one form of the invention by manual operations and calculations, and in another form of the invention a decimal point dial is automatically operated in response to movements of the slider and rider of the slide rule during the course of a calculation to thereby assist in determining the decimal point.

This application is a continuation-impart of my copending application, Serial No. 451,449, filed July 18, 1942, now abandoned.

One object of my invention is to provide a decimal point locator which may be conveniently arranged on the back of a slide rule and which is provided with indicia arranged to assist in indicating the decimal point location in the answer to a problem in response to the calculation of such problem on a reduced scale version of the slide rule printed on the back thereof in a plurality of sections to indicate directly any answer that goes on scale.

Another object is to simplify the determination of the decimal point in the answer so that calculation, according to a simplified formula involving the number of* factors, digits and zeros in the calculation, will indicate a Diierence Number, which Difference Number can then be utilized for the purpose of registry with a certain Region Number, the Region Number being found by the number of times the answer goes oif scale, as determined by manipulations of the reduced plurality of slide rule scales on the back of the slide rule.

Another object is to provide certain scales for a Diiference Number and a Region Number which cooperate with each other to indicate the position of the decimal point in numbers of the first, onehalf and onefthird powers of the numbers. These scales may be extended on to additional powers of the numbers in an obvious manner.

A further object is to provide one form of the invention wherein the Region Number is automatically determined by mounting the Region Number scale and a dial on the rider of the slide rule, the dial being rotated through a gearing arrangement from racks on the stator and slide of the slide rulek with which certain pinions of the gearing mechanism cooperate, thereby operating the dial in response to movement of either the slide or the rider in relation to the stator.

Still a further object is to provide a Difference Number scale on lthe dial and utilize the Region Number scale also for reading digits and zeros in the answer, the Region Number scale being thereby a combined Region Number and Digits-Zero scale on which the Digits and Zeros in the answer can be read.

An additional object is to provide an automatic Region Number, Difference Number and Digits- Zero indicator which is readily applicable to a rotary slide rule.

With these and other objects in view, my invention consists in the construction, arrangement and combination of the various parts of my decimal point locator and its relation to a slide rule whereby the objects contemplated are attained, as hereinafter more fully set forth, pointed out in the claims and illustrated in the accompanying drawings, wherein:

yrule embodying my invention showing the left half of the rule;

Fig. 1A is a similar view showing the right half of the rule;

Fig. 2 is a similar view showing the left half of the back of the rule on which the decimal locator is mounted;

Fig. 2A is a similar view of the right half of the back of the rule;

Fig. 3 is an elevation of a special type of rider having the decimal point locator mounted thereon and geared to the stator and slide of the slide rule;

Fig. 4 is an enlarged sectional view on the line 4 4 of Fig. 3 showing the gearing arrangement and associated parts;

Fig. 5 is a sectional view of a rotary slide rule to which the invention has been applied; and

Fig. 6 is an enlarged sectional View on the line 6--6 of Fig. 5.

On the accompanying drawings I have used the reference numeral I0 to indicate the stator oi my slide rule; I2 the slide thereof; and I4 the rider. The rider I4 is transparent and has the usual hairline I6 thereon for indicating purposes.

Referring to the front of the slide rule, as shown in Figs. l and lA, there is a succession of scales from top to bottom thereof. The rst four scales are on the stator, three of them being cube root scales indicated V. These may be referred to as the upper, middle and lower cube root scales without the necessity of applying reference numerals to them. The fourth scale is a log scale, indicated as L, and its purpose will be referred to hereinafter.

The next five scales are' on the slide and appear through a sight opening I3 of the stator I0. These iive scales may be classified as follows: Upper T scale (indicated T) is the tangent scale of angles between 5.7 and 45. The lower T scale (also indicated T) is used on angles between 45 and 84.3. The third scale on the slide (ST) is for sines and tangents of small angles, as will be hereinafter explained. The fourth scale (S) is for .eines of angles between 5.73 and 90. The fth scale is the regular C scale of an ordinary slide rule used in multiplication and division computations.

The next scale (D) is the ordinary D scale on the stator of the slide rule used in multiplication and division. computations. Below the D scale are an upper square root scale and a lower square root scale (also indicated These will be hereinafter referred to as the upper scale and the lower scale.

In Figs. 2 and 2A the back of the slide rule is shown. The back is a combined rough result calculator and decimal point locator, and has the C, D, T, T, ST and S scales of the front of the rule also printed thereon and designated, respectively, C, D', T', T', ST and S. The C scale is arranged on the stator l in this instance, and the D scale on the slide I2. The D', T', T', ST and S scales appear through a sight opening l in the stator I0, portions of the stator adjacent the ends of the opening being broken away to show instructional data on the slide l2.

The C scale is condensed in length with respect to the C scale on the front of the slide rule, and is repeated four times. The D', T', T', ST' and S' scales are repeated twice on the slide, once at the left for multiplication and once at the right for division.

On the stator I0 above the C scales are two Region scales (see Figs. 2 and 2A). Reading from right to left, the upper region scale has a -l, a 2, a 3 and part of a -4 region indicated. Reading from left to right, the lower region scale has a 0 region, a +1 region, a +2 region, a +3 region and part of a +4 region indicated. These regions are substantially coextensive with the repeated C scales.

In determining region numbers the left hand stator scale C is considered on-scale with a region number of 0. The three stator scales C to the right of the left hand scale are considered off scale to the right regions and are designated respectively +1, +2 and +3. This is particularly for problems in multiplication.

In division, readings extend off scale to the left. The right hand C scale of the stator is considered on-scale with a region number of 0 again and the three stator scales to the left of the right hand one are considered as off scale to the left regions +1, -2 and 3. If any problem extends beyond the length of four stator scales, higher plus and lower minus region numbers are used but have to be kept track of.

On the slide l2, and appearing through the sight opening l5, is a Difference Number scale indicated DN. This scale at the center has a 0 with minus numbers to the left of the 0 and plus numbers to the right of the 0. A black arrow, indicated at i3 in Fig. 2, is placed over the -1. Several other notations are preferably provided on the slide I2, as shown in the drawing for convenience in indicating the operation of the decimal point locator to the user, but will not be referred to in detail.

On the stator l0 below the scale DN of the slide there is a scale indicated as RN-DZ. This is the Region Nuinber-Digit-,Zero scale, and below 4 it are further RN-DZ scales for square root and cube root The foregoing description is somewhat brief and does not explain the purpose of the various scales and arrangements, Their purpose is believed best explained by rst explaining the theory of decimal point location with a few problems in both multiplication and division and a process of algebraic substraction on the back of the slide rule followed by an explanation of the use of special scales for square roots and squares, cube roots and cubes, logarithms, trigonometry, and computations involving Several scales. I shall then go into the theory of the decimal point locator and the formula which I use as a basis for its operation.

DECIMAL POINT LOCATION -LocA'rING THE DECIMAL POINT 1N MULTIPLICATION Although in simple problems the position of the decimal point can be estimated, there are many problems in which this part of the work is more diiilcult. There are several ways of simplifying the work. The method described below makes use of the decimal point locator already described on the back of the slide rule.

When numbers are greater than 1 the number of digits to the left of the decimal point will be counted. Thus '734.05 will be said to have 3 digits. Although as written the number indicates accuracy to live digits, only three of these are at the left of the decimal point.

Numbers that are less than 1 may be written as decimal fractions. Only positive real numbers are being considered in this discussion. Thus .673, or six-hundred-seventy-three thousandths, is a decimal fraction. Another example is .000465. In this number three zeros are Written to show where the decimal point is located. One way to describe such a number is to tell how many zeros are Written to the right of the decimal point before the iirst non-zero digit occurs.

In scientific work a zero is often written to the left of the decimal point, as in 0.00541. This shows that the number in the units place is detinitely 0, and that no digits have been carelessly omitted in Writing or printing. The zeros will not be counted unless than are (a) at the right of the decimal point, (b) before or at the left of the first non-zero digit, and (c) are not between other digits. The number 0.000408 will be said to have 3 zeros (that is. the number of zeros between the decimal point and the 4).

Examples Number of Number of Number digits to zeros to bc be counted counted provides a rough checkv of the result. In the space just above the C scale the hairline falls within a region marked +1. This is the Region Number for this multiplication. The settings may be made very quickly and roughly, since only a. check and the Region Number are desired.

Note that there are two D scales on the slideone at the left end and one at the right end. The left-hand D scale is used for multiplication, starting at the left and reading toward the right. The region number is read on the bottom region number scale (just above the C scale), as 0, or +1, or +2, etc.

Set the lower point of the arrow I8 above the "Region Number on the RN-DZ scale. In this example, the arrow is set above +1.

Now in the example 3420000 .000694, the number 3,420,000 has seven digits. The number .000,694 has 3 zeros. The product of 2 numbers or factors is to be found. The sum of the number of zeros (3) and the number offactors (2) is 5. This 5 subtracted from the number of digits (7) leaves +2. Now locate +2 on the DN scale, and below it read +4 on the RN-DZ scale. This +4 means that the product has 4 digits, and hence the result is approximately 2370.

Although it is diflicult to describe this process briefly, with a little practice results can be obtained very quickly by the following rules.

Step 1.-Use rough settings of the numbers on the C and D scales to nd the Region Number in the region above the answer on the C scale. The slide I2 projects to the right, and the Region Number is in the lower or regions. In a very few computations of continued products it may happen that the Region Number is larger than +3. When this occurs the left-hand D scale extends beyond the rule at the right end. By use of the hairline, the last product obtainable on the rule should be moved to the left-hand C scale. The multiplication is then continued, but +3 must be added to the Region Number found. Thus the Region Number for 9 9 9 9 9 9, or 96, is (+2 +3), or +5. Similarly, another +3 would be added if the result falls oir the rule a second time, etc. Thus the Region Number for 99 is +8.

Step 2. Set the point ofthe arrow i8 over the Region Number on the RN-DZ scale.

Step 3,- C'ount the number of factors to be multiplied andthe number of their zeros (if any), and subtract the total from the number of digits.` Locate this result on the DN scale, and read the number below it on the RN-DZ scale.

If this number has a sign, it is the number of digits in the product.

lif this number has a sign, as 1, 2, 3, it means that the product has zeros to the right of the decimal point, and the numerical value', as l, 2, 3, tells how many of these Zeros there are.

If this number is zero, the decimal point is to the left oi the product. (That is, the product has no digits and no zeros.)

Step 4.-Point oi the decimal place in the result.

Example-43X0.0000306 0.28.

First ignore the decimal points and multiply 43 306 28 on the C and D scales. The rst three gures of the result are 368. Next, repeat the multiplication on the C and D scales as a check and as a means of finding the Region Number (+1). Set the arrow i8 over +1 on the RN- DZ scale. New count the number of factors (3) and the number of zeroes (4) in the problem. The sum (7) when subtracted from the number of digits (2) leaves 5. Below 5 on the DN scale Number RN is found to be 1.

6Z locate fi-on the RN-DZ scale. Therefore the result has 3 zeros to the right'l of the decimal point. The result, then, is .000368, approximately.

LOCATING THE DECIMAL POINT IN DIVISION In'- locatirig the decimal' pointv of the result after a division', an additional process is necessary in Step- 3 above'. It will usually be best to write the computation i'r'i the form of a fraction as in the exam-ple below. Then Step 3 above is done for both the numerator (dividend) and the denominator (divisor) of the fraction. In each case either a positive or a negative number is obtained. Finally, the number obtained for the denominator must be subtracted algebraically from the number obtained for the numerator. This result will be called the Difference Number or DN, and it may be located on the DN scale. Thereafter the steps are the same as before. Start the divi-sion on thel left-hand D and C scales. The Region Number' is read above the left index of a'" D scale. Ii the quotient falls on the C scale, the Region Number is zero. If the quotient falls off the C scale (to the left), it and the Region Number may be read above the left index oi the D scale at the right end of the slide.

For combi-ned multiplication and division, a more general rule is needed. If the left-hand D scale can be used in reading the result, the Region Number is read in one of the lower Re gions as 0, or l, or 2, etc. At the left end 0n the back of the slide I2 a notation, Left Slider Scales Refer to Regions, serves as a reminder of this fact (Fig. 2). y

If the right-hand D scale must be used in reading the result, the Region Number is read from the upper or Regions as 1, or 2, er' 3, etc. At the right end of the slide there is a reminder of this fact (Fig. 2A).

Ensamble-Divide .0035x726 by 9200 .0'000064.

Step l: Form the fraction .0035x726 ezooxnooooci Step 2: Using the C and D scales, compute The first three lgures of the result are 432.

Step 3: Repeat, using the C scale and the D' scale at the left end of the rule. The Region The quotient gures' 43 may be readily checked by a glance at the C scale.

Step 4: (a) Count the number of factors (2) and the number of zeros (2) in the numerator of the original fraction. When the total (4) is subtracted from the number of digits (3), the result is +1; that is, 3 4= 1. (b) Count the number of factors (2) and the number of zeros (5) in the denominator. When the total ('7) is subtracted algebraically from the number of digits (4), the result is 3; that is, 4 7=-3. (c) Subtract the result for the denominator from the result for the numerator; i. e., 1 3) :+2. That is, subtract algebraically:

This +2 is the Difference Number. DN.

asoasss 7 Steps 4a, 4b and'4c can be concisely stated numerically thus:

Step 5: Set the arrow |'8 over the Region Number 1) on the RN scale. Locate the Difference Number (+2) on the DN scale. Read the number of digits (+2) on the DZ scale.

Step 6: Place the decimal point so that the product (see Step 2) has two digits; the product is 43.2.

ALGEBRAIC SUBTRACTION oN THE BACK F THE SLIDE RULE Some of the subtractions required in Step 4 above involve the use of negative numbers, such as 1, 2, 3, 4, etc. Students who are familiar with these numbers will have no difficulty, but it is a good idea to jot down the numbers as above, following Step 4. (If a number has no or sign before it, a sign may be put there without changing the numerical value.)

On my slide rule these subtractions can be done (or checked) mechanically. The DN and RN scales are used. The method is as follows:

Rule- Locate the number to be subtracted on the DN scale, and set it overthe number it is to subtracted from, located on the RN scale. Read the result on the RN scale under the 0 of the DN scale.

Examples (a) 4 6=? or subtract Locate +6 on the DN scale (use indicator), and set it over +4 on the RN scale. Under 0 of the DN scale nd the result, 2. (b) -3-5=? or subtract Locate 5 on the DN scale, and set it over -3 of the RN scale. Under 0 of the DN scale nd the result, -8.

(c) -1-(-2)=? or subtract Locate -2 on the DN scale, and set i-t over 1 of the RN scale. Under 0 of the DN scale nd the result, +1,

(d) 7-(-5)=? or subtract Locate -5 on the DN scale, and set it over +7 of the RN scale. Under 0 of the DN scale find the result, +12.

USE OF SPECIAL SCALES SQUARE ROOTS AND SQUARES When a number is multiplied by itself the result is called the square of the number. Thus 25 or 5X5 is the square of 5. The factor 5 is called the square root of 25. Similarly, since 12.25=3.5 3.5, the number 12.25 is called the square of 3.5; also 3.5 is called the square root of 12.25. Squares and square roots are easily found on a slide rule.

Square root-Just below the D scale is another scale marked with the square root symbol,

Rule-The square root of any number located on the D scale is found directly below it on the scale.

EampZes.-Find V4. Place the hairline oi' the indicator over 4 on the D scale. The square root, 2, is read directly below. Similarly, the square root of 9 (or \/9), is 3, found on the scale directly below the 9 on the D scale.

Reading the Scalea-The square root scale directly below the D scale is an enlargement of the D scale itself. The D scale has been stretched to double its former length. Because of this the square root scale seems to be cut 01T or to end with the square root of l0, which is about 3.16 To nd the square root of numbers greater than 10 the bottom scale is used. This is really the rest of the stretched D scale. The small iigure 2 near the left end is placed beside the mark for 3.2, and the number 4 is found. nearly two inches farther to the right. In fact, if 16 is located on the D scale, the square root of 16, or 4, is directly below it on the bottom scale of the rule.

In general, the square root of a number between l and 10 is `found on the upper square root scale. The square root of a number between 10 and is found on the lower square root scale. If the number has an odd number of digits or zeros (1, 3, 5, 7, the upper scale is used. If the number has an even number of digits or zeros (2, 4, 6, 3, the lower V' scale is used.

On the slide rule, the first three (or in some cases even four) heures of a number may be set on the D scale, and the rst three (or four) iigures of the square root are read directly from the proper square root scale.

On the back and near the bottom of the rule is a scale (marked useful in finding the number of digits (or zeros) in a square root. Locate the number oi digits on the RN scale. Read the number of digits in the square root on the scale marked just below. The negative or minus signs indicate zeros in the number or in the square root.

Examples (a) Find \/248. Set the hairline on 248 of the D scale. This number has 3 (an odd number) digits. Therefore the gures in the square root are read from the upper scale as 1575. Under the +3 of the RN scale we nd +2 of the scale. Hence the result has 2 digits, and is 15.75 approximately.

(b) Find \/563000. Set the hairline on 563 of the D scale. The number has 6 (an even number) digits. Read the gures of the square root on the bottom scale as 75. Under +6 of the RN scale nd +3 on the scale. The square roo-t has 3 digits and is '750 approximately.

(c) Find \/.00001362. Set the hairline on 1362 of the D scale. The number of zeros is 4 (an even number). Read the iigures 369 on the bottom scale. Locate -4 on the RN scale. Find 2 under it on the y scale. The result, then, has 2 zeros, and is .00369.

Squaring is the opposite of iinding the square root. Locate the number on the proper bottom scale (marked and with the aid of the hairline read the square on the D scale.

Eamples (l1) Find {62800)2. Locate 628 on .the bottom V scale. Find394 above `it on the D scale. The number has digits. Locate +5 on the scale on the back of the rule. There Aare two +5s and above lthem on the 'RN scale We nd +9 and +10. Hencethe square has either 9 or 10 digits. Since, however, .628 was located on the bottom scale, the square has fthe even number of digits, or 10. Theresult is 3,940,000,000.

(c) IFind (.000254)2. On the D scale read 645 above Ythe 254 of the upper scale. The number has 3 zeros. On the back of the rule locate Vthe two -3s on the scale. Above them find -6 and 7. vSince 254 was located on the scale for cdd zero numbers, the result has 7 zeros, and is .0000000645 CUBE ROOTS AND CUBEs At the top of the slide rule-on the front (Figs. 1 and 1A) there is a cube root scale marked It isa D scale which has been stretched to three times its former length, andthen cut into three parts which are printed one below the other.

Rule-The cube root-of any number on the D scalefis found directly above it on the scales.

At'the left end of the cube root scales a small table serves as a guide as to which scale to use. Also, on the bottomof the Vback of the rule is a f/ scale for determining the number of digits orzeros. It is used in exactly the same .way as the scale above it was used.

Examples (a) Find Vif Set the hairline I6 over the 8 ofthe D'scale. On the topmost scale of the ruleread 2 under the hairline.

(b) Find i/ 27. Set the hairline over 27 of the the D scale. On the middle scale, nd 3 under the hairline.

(c) Find i/ 372. Set the hairline over`372 oi the D scale. On the bottom scale rind 719, or 7.19,

Cubing is the opposite of finding thefcube root.

Rule-The cube of any number located on the scale' is found directly below it on the D scale.

Example-4a) Find (3`2.8)3. Locate 32.8 on the middle scale. On the D scale read directly below it the figures of the cube 353. Since 32.8 is a two digit number, aglance at the +2 of the scale on the back of the rule shows that it falls under +4, +5,.or +6 of the RN scale. But since 328 is found on the middle scale, the number of digits is 5. The result is-35300 approximately.

LoGARITHMs The L scale on the front'of 'the stator l0 just above the slide l2 is used for finding the logarithm (to the base I0) of any number.

Rule-Locate the number on the D scale, and read the mantissa of its logarithm (to base l0) directly above it on the L scale.

Example-Find log 425. Set the hairline over 425 on the D scale. Read:the mantissa of the logarithm (.628) onthe L-scale. Since the number 425 has three digits, the .characteristic is 2 and the logarithm is 2.623. As an aid to the memory, the rule for finding the characteristic isprinted on the back of the slide I2.

'If the logarithmyof a number is known, the number itself may be found by reversing the above process.

Example- If log :.r=3'.248, nd Set the hairline over 248 of the Lseale. Below it read the number 177 "on'the Dlscale. Then 93:1770

approximately.

l0 EampZe.-Find log .000627. Opposite 627 on the D scale find .797 on the L scale. Since the number has 3 zeros, thevcharacteristic is 4, and the logarithm is usually written 6.797-10, or 0.797-4 TRIGONOMETRY Sines and cosines The scale S on the front of the slide i2 is used in iinding the approximate sine or cosine of any angle between 573 and 90. Since sin =cos (-:c) the same graduations serve for both sines and cosines. Thus sin 5=cos (90-6)=cos 84. The numbers at the right of the longer graduations are read when sines are to be found. Those at the left are used when cosines are to be found. `Angles are divided decimally instead of into minutes and seconds. Thus sin 12.7 is represented by the 7th small graduation to the right of the graduation marked 78112.

Ruler-Tc find the sine or cosine of an angle on the S scale, set the hairline i6 on the graduation which represents the angle. Read the sine on the C scale under the hairline. If the slide is placed so the C and D scales are exactly together,.the mantissa of the logarithm of the sine (log sin) may also be read on the L scale.

Examples (a) Find sin :r: and also log sin :c when a1=l5 30. Set left index of C scale over left index of Dscale. Set hairline on 15.5 (i. e., 15,30) Read sin :1::.267 on the C scale. Read .427 on the L scale. Then the log sin 35:9.427-10.

(b) Find cos a: and log cos x when zc=42 l5 (or :r=42.25). ,Observe that -the cosine scale decreases from left to right, or increases from right to left. Set the hairline over 42.25 on the S scale (reading from the right). Find cos 4225:.740 on C scale. Find .869 on L scale. Hence log cos 42 l5'=9.869-'10,

Tangente The upper T scale on the front of the slide I2 is used to vfind tangents of angles between 5.70 and 45. These tangent ratios are all between 0.1 and l; that is, the decimal point is at the left of the number as read from the scale. The lower T scale is used in finding tangents of angles between 45 and 84.3. These tangents are between 1 and 10; kthat is, theyall have one digit to the left of the 'decimal point.

Rule-Set the angle value on the graduation which represents the angle and read the tangent on the C scale. If the C Vand D scales have their indices exactly together, the mantissa of the logarithm of the tangent may also be read on the L scale.

Examples (a) Find tan :v and log tan a: when m=9 50. First note that of 1 degree=.83, approximately. Hence Set the left index of the C scale and of the D scale opposite each other. Locate :9.83 on the upper T scale. Read ltan :c=.l73 on the C scale, and read .239 on Vthe L scale. Then log :tan :10:9.239-10.

(b) Find tan .r When :c:68.6. Use the lower T scale. Read 255 on `the C scale. Since all angles on the lower T scale have tangents greater than 1 l1 (that is, have one digit as defined above), tan 11522.55.

Sines and tangente of small angles The sine and the tangent of angles -of less than about 5.7 are so nearly equal that a single scale `on :the 'front of the slide I2, marked ST, may be used for both. The graduation for 1 is marked with :the degree symbol To the left of it the primary graduations represent tenths of a degree. The graduation for 2 is just above the graduation for 35 on the C scale. The graduations for 1.5 and 2.5 are also numbered.

A small scale l1 on the back lof the rule at the center f the slide I2 shows the number of zeros in the sine of angles between 0 and 90, and the number of zeros or digits in the tangents of most of these angles. Sines or tangents of angles on the ST scale have one zero. Sines (or c-osines) of all angles on the S -scale have no digits or zeros-the decimal point is at the left of gures read from the C (or D) scale. All angles located on the upper T scale `also have the decimal `point of the tangents at the left of the numbers. Angles located on the lower T scale have one digit in :their tangents. Tangents of angles larger than 84.3 are not read from the rule; they increase rapidly `and have :at least two digits.

Two seldom used special graduations are also placed on the ST scale. One is marked with the symbol for minutes `of angle, and is found just to the left of the graduation for 2. When this graduation is set opposite any number of minutes on the D scale, the sine (or the tangent) of an 'angle of that many minutes may be read on the D scale under the C index.

Sine 0=0, and sin 1=.00029, and for small angles ythe sine increases -by .00029 for each increase of 1' in the angle. Thus sin 2=.00058; sin 3.44=.00100, `and the sines 0f all angles between 3.44 and 34.4 have two zeros. Sines of angles between 34.4 and 344 (or 5.73) have one zero. The tangents of these small angles .are very nearly equal 4to the sines.

Example- Find sin 6'. With the hairline set the minute graduation `opposite 6 located on the D scale. Read 175 on the D scale under the C index. Then sin 6=.00175.

The second special graduation is marked with 'the symbol for the seconds of angle and is located near the graduation for 1.2. It is -used in exactly :the same way as the graduation for minutes. Sin 1":.0000048 approximately, and the sine increases by this amount for each increase of 1 in the angle, reaching .00029 for sin 60" or sin 1'.

CoMPU'rATIoNs INvoLvING SEVERAL SCALES Many calculations are simplified by using several different scales. Suppose it is necessary to compute the areas of many circles. Since the formula A=1rr2 can be written as a proportion, that is,

the following rule `will hold. Set vr of the C scale opposite the index of the D scale. Locate the hairline over the value of the radius r on the scale. Read the area under the hairline on the C scale. If the diameters are known, instead of the radii, then or A:.7854d2. Hence set .785 on the C scale opposite the index (usually right-hand) of the D scale. Locate the hairline over the value of the diameter on the scale, and read the area under the hairline on the C scale. In similar fashion, but using the cube root scales, the volume of spheres may readily be found.

Many formulas involve both trigonometric rartios and other factors. By using several difierent scales such computations Iare easily done.

Example- Find the length of the legs of a right triangle in which the hypotenuse is 48.3 ft. and one acute angle is 25 20.

The side opposite the given acute angle is equal to 48.3 sin 25 20. Hence we compute 48.3 sin 25.3. Set the index (right-hand index in this example) of the C scale on 48.3 of the D scale. Move the hairline over 25.3 on the S scale. Read 20.7 under the hairline on the D scale. Another method is to set the left index of the C scale and D scale opposite each other. Set the hairline over 25.3 on the S scale. Move the slide so that (right) index of the C scale is under the hairline. Read 20.7 on the D scale under 48.3 of the C scale. The length of the other leg is equal to 48.3 cos 25.3 or 48.3 sin 64.7=43.7,

The decimal point in this result may be found by use of the special point-locator scales on the back of the rule. Set 48 on the left-hand C' scale, and locate the hairline over sin 25 on the small S scale of the slide. The Region Number (RN) found under the hairline is +1. In this example when the arrow i8 is set over -i-l, the number under O is +2 on the DZ scale, showing that the result (20.7), has two digits.

Powers involving the fractional exponents 2/ 3 and 3/2, or in other words, combinations of squares of cube roots, and of cubes of square roots, may be done with one setting of the hairline.

Rule to compute :z2/3: Set a on the scale.

Read ft2/3 on the scale.

Rule to compute cl3/2: Set a on the scale.

Read :z3/2 on the scale.

Eample.-Find the surface area of a cube which has a volume of 64 cu. in. Since V=e3, then c=/V=V1/3. Also S=6e2 or S=6V2/3. If V:64, then S=6 642/3. To find 642/3, set the hairline over 64 on the scale. Read on the scale, 642/3=16.

Example-A formula sometimes used in aeronautical computations is It is used to help answer questions like the following: If the weight of a plane is increased 15%, what effect has this on the required horsepower Pn- Solution: In this case Wn -WE- 1.15

Then (1.15)-"/2 must be computed. Set 1.15 on the /scale. Read 1.23 on the l/ scale. Hence the horsepower must be increased by 23%.

Example-In radio theory, the resonance frequency is given by the formula l f "2m/m Find f when L=253 microhenries and C= micro-micro farads. Then 1 J 21m/.000,253 .oodooobouoco For slide rule computation it is more convenient to write this in the equivalent form:

1r) on the D' scale under the hairline, and then move the hairline to the left index. The new Region Number is +2. Continue in this way until all the remaining factors in the denominator (i. e., 1r, 253, and 90) have been used. The final RN=+3.

For the numerator D (F+Z) :0. For the denominator D-(F+Z) =3-(5+13) 15. Hence DN=-(-15)=15. With the arrow I8 over R'N=-3, note 13 on DZ under the DN(15). At

the same time note 7 on the scale below the 13. There are 13 digits in the number under the square root symbol, and 7 digits in the final answer.

Now repeat the calculations using the C and D scales. After the last division the square root 'of the result is found on the upper scale on 'ther front of the rule, since the number has 13 (an odd number) digits. The result is 1,060,000 and the frequency is about 1060 kilocycles.

Ensamble- In the study of meteorology it is sometimes desirable to compute where p=density of the air, w=anguiar velocity of the learths rotation, =latitude, and u=co efficient of eddy viscosity. Find a, when p=0.0011, w=0.0000729, (eL-40, and u=116.

The ease with which such a calculation can be done on my slide rule is shown below. We have to compute:

Usingv the compressed scales C', D', S at the left end of the decimal point locator on the back of the slide rule, find RN=+1. Set the arrow I8;- over +1. On the DZ scale under -11 read 9. Gn the scale under -11 read 4. Then the. result has four zeros.

Next, set 116 on C scale over 11 on D scale, move runner to 729 on C scale. Move right index under the rider I4; then move the hairline i6v over sin 40 on S scale. The. number under the square rootsymbol has 9 (anodd number) zeros. Hence readthe result on the upper square root scale, as: 2111. Point off the decimal place in the result m0000211'.

Other values', such. as log a could be read from the same settingv but would not usually be found in the' example: log :5.f.648-1.0.

Under the leftmost index of THEORY or THE DECIMAL POINT LOCATOR A brief discussion of the theory of the special features of my decimal point locator follows. A knowledge of the theory of logarithms is required for a complete understanding of how the decil mal point locator achieves its results.

When multiplications are carried out by means of logarithms it frequently happens that the sum of two or more of the mantissas exceeds 1. vIn ordinary computation this results in a carry number which is added to the Characteristic. In slide rule computation this situation corresponds to cases in which the slide extends too far to the right and the other index is used in order to read the result of a multiplication on the D scale. On my decimal point locator the Positive Regions above the small C' and D scales represent carry numbers of +1, +2, etc., resulting from the vaccumulation of mantissas. The Negative Regions +1, +2, etc., represent the reduction (borrowing) in value of the characteristic which occurs when, in division by logarithms, the mantissa subtracted exceeds the mantissa of the dividend or numerator.

Examples (a) Multiply 36 7700.

By logarithms; log 36:1.556 log 77=3.886

Sum=5A42 Answer 276000 Observe that the sum of the characteristics, 1+3=4, is increased by 1, the carry number from the mantissas. When the multiplication is done on the slide rule, the addition of the mane tissa is accomplished by means of the adjacent scales. When the C and D scales are used, the Region Number (or "carry number) is seen to be +1.

(b) Divide 231+'68.

By logarithme log 231:2.364 log 68= 1.833

Diierence=0-531 Answer= 3.40

Observe that it is necessary to borrow 1 from from the 2 in the characteristic in order to subtract. When the division is done on the slide rule, the subtraction of the logarithms is accomplished by means of the adjacent scales. When the C and right hand D' scales are used, the Region Number (or amount borrowed) is indicated by 1. Thus the Regions are a convenientA mechanical device for keeping track of the gain or loss in Value of the characteristic as a result of carrying (or borrowing) -in operating with the mantissas.

Consider now a calculation requiring the product of'F factors; e. g., A1 Az As AF. If A1 contains any number of digits (including zero), let d1 represent this number. In general, if Ai contains any number of digits (including zero) let di represent the number of these digits. Simllarly, if Ai contains any zeros let ai represent the number of these zeros. Suppose now that any contribution to the characteristic carried from the sum oi the mantissas is ignored. Then the uncorrected characteristic of the product above is where the subscripts k, l, p, q, r, etc.y represent 15 integers of the set 1, 2, 3, F and are such that every integer of this set is represented once. Under these conditions when the parentheses in this sum are removed, -1 will occur F times and .the sum of these terms will be -F. Then Represent the sum of the number of digits in the factors by D, and the sum of the number of zeros by Z. Then Then C=D- (Z-i-F) which indicates that the uncorrected characteristic cf the product of F factors may be found by subtracting the sum of the number of zeros and the number of factors from the number of digits.

When an expression to be computed has the form the same rule may be used to calculate the characteristic C of the denominator. Then the characteristic of the quotient is, by the usual rules, C-C. Since in general C and C may be positive, negative, or zero, algebraic subtraction must be used throughout. It must be remembered that the characteristic so computed must be corrected by the addition of numbers resulting from any carrying or borrowing in operating with the mantissas which affects the characteristic.

In the notation used on the back of the slide rule, the Difference C'-C=DN. When no division (except by l) is called for, C:0, and the Difference Number is merely D-(F-l-Z) calculated for the numerator. The correction to DN is made mechanically by setting the arrow I8 over the Region Number in the RN scale, and finding the number of digits (or zeros) in the result under the DN number on the DZ scale. This is, of course, merely simple algebraic addition by means of uniform (rather than logarithmic) scales.

If the arrow I8 had been placed over the O of the DN scale, the usual rules, such as the number of digits in the result is one more than the characteristic would have applied. The calculation of the one more in this rule is taken care of mechanically by placing the arrow I8 over -l instead of 0. A similar sort of discussion applies to the case in which the result has zeros to the right of the decimal point.

An alternative rule useful in locating the decimal point in division may be derived as follows: Let subscripts n and d indicate the numerator and the denominator respectively. Then According to this rule one may compute DN as follows:

(l) Count the number of digits in numerator and subtract the number of digits in the denominator.

(2) Repeat for factors.

` (3) Repeat for zeros.

(4) Add the results of steps (2) and (3) and subtract the sum from the result of (1).

The four steps of the rule just described may be simplied into the following formula, taking into consideration that l is to be subtracted from the calculation of the digits, factors and zeros in the numerator and denominator to get an accurate answer:

With the arrow I8 placed over -l on the DN scale of the slide, however, the -1 in the above formula can be omitted, thus producing the formula:

to be used in connection with the calculation of the decimal point when using the decimal point locator illustrated on the back of the slide rule.

In order to eliminate the necessity of having to roughly perform the calculation on the back of the slide rule (which has already been performed accurately on the front of the slide rule to get the number in the answer), a structure as shown in Figs. 3 and 4 may be provided. The stator II) has a rack 26 secured along one of its edges. The slide I2 has an extension 22 through a slot 24 of the stator which terminates along its edge in a rack 26. The rider I4 has a transparent part |41 through which the scale divisions for the C and D scales, etc., on the front of the rule can be seen, and this part of course would be provided with a hairline in the usual manner. A shaft 38 has its ends secured to the rider I4. Rotatable on the shaft 30 is a sleeve 32 having secured thereto a pair of pinions 34 and 36. The pinion 34 meshes with the rack 26, and a gear 38 meshes with the pinion 36.

The gear 38 is freely rotatable on a shaft 40 and has a pinion 42 integral therewith. The pinion 42 meshes with a gear 44 having a pinion 0 thereof bevel gear teeth 54.

Meshing with the rack 2U is a pinion 56, and integral therewith is a second pinion 58. The pinions 56 and 58 rotate freely on the sleeve 32. A gear 60 is freely rotatable on the shaft 40 and has integral therewith a pinion 62 meshing with a gear 64. The gear 64 is secured to the sleeve 48 and also to the sleeve is secured a pinion 66. The pinion 66 meshes with a gear 68 on the face of which a dial l0 is printed. This dial is on the back side of the rider I4 opposite the hairline on the front of the rider.

Freely rotatable on the shaft 40 is a second bevel gear l2 opposite the bevel gear 54. The bevel gear 'I2 has frictionally engaged therewith a dial 14. A spring washer 96 0f the cupped disk type effects such engagement, the bevel gear being backed by a shoulder 95 on the shaft 40. Freely rotatable on the shaft 40 between the bevel gears 54 `and 'I2 is a disk I6 carrying, preferably, three bevel pinions 'I8 meshing with the bevel gears 54 and 12. The elements 54, 'I2 and I8 may be broadly termed as differential gearing.

A lock is provided for the rider I4 in relation to the stator and for the disk 'I6 during certain operations of the slide rule. This lock consists of a lock plate having flanges 82 and 84 frictionally coactible respectively with an annular groove 86 between the pinions 56 and 58 and the periphery of the disk 16. The lock plate is slidably mounted as between a guide flange 8I and an adjacent portion of the rider I4 and has a slot 83 to clear the pin 30. The lock plate 80 may be lowered to locked position by a pair of cams 88, as shown in Fig. 4, when one finger-piece 92 locking ringer 8i) being of the cams 88 is presesed toward the rider. The locks at v82 and 84 may be released from the Igroove 86 and the disk 'I6 -by pressing a second linger-piece 90 of the cams 88. Each cam 88 has high and low flats 88a and 88b coacting with the lock plate 80 for this purpose. The cams are pivoted on a pivot pin 94 which is supported by a boss 95 on top of the rider I4.

, The dial 'I4 has arranged thereon the DN scale shown in Figs, 2 and 2A, the arrangement, however, being in a circle instead of straight, This dial also has the arrow I8 applied thereto over the +1, as in the DN scale of Figs. 2 and 2A. The dial 'I0 has applied thereto the RN -DZ scales for the first power numbers, square root numbers and cube root numbers 3/ as on the stator IIJ in Figs. 2 and 2A below the sight opening I5. The number of divisions in these scales determines the gear ratio :between the dials and the racks 28 and 26, the ratio being such that when the rider traverses the slide rule from the left index to the right index thereof the dials move in relation to each other the distance of one scale division, as will be explained in the examples which follow.

During the course of a calculation, for example 2 3 3, we begin with the indices or 1 and marks of the slide I2 corresponding to the indices of the stator I8 and the hairline I6 of the rider I4 over the left indices. The dial 'I4 is set with its arrow i8 of the DN scale matching the 0 of the RN-DZ scale on the dial 18. The first operation then is to move the rider until the hairline matches the 2 on the D scale of the slide rule. This results in rotation of the pinions 34 and 56 in the same direction (clockwise) and at the same speed with the pinion 56 rotating the elements 86, 58, 60, 62, 64, 48, 66, 68 and 10. The dial 78 turns counterclockwise, considering rotation from the front of the slide rule (or clockwise, considering it from the back, Fig. 3), a distance corresponding to the logarithm of 2. At the same time, the elements which are rotated by the pinion 34 are 32, 36, 38, 42, 44, 46, 52 and 54. The bevel gear 54 rotates counter-clockwise and in so doing it rotates the disk 'I6 also counter-clockwise but at half the speed of the bevel gear because at this time the disk is unlocked by the unpressed downwardly. Consequently, the dial. 'I4 remains stationary, due to its frictional engagement with the shoulder 95 of the shaft 40 under the action of the friction Washer, and the arrow I8 thereby still indicates the 0 region of the RN-DZ scale.

The next step in the calculation is to move the slide I2 to the right until its left index is under the hairline I6, and this results in no movement of either` the dial 'I0 or the dial 74, since the rack 26 of the slide now rotates the bevel gear 54 (this time clockwise) through the intermediate elements 34, 32, 36, 38, 42, 44, 46 and 52, and the disk 16 rotates at half the speed of the bevel gear with no movement being imparted to the dial 14.

The next step in the operation is to move the rider until the hairline is over 3 on the C scale, which moves the dial I0 again, as already explained in the rst step of the operation, this time advancing it clockwise (from the back, Fig. 3) until the logarithm of 6 on the RN-DZ scale corresponds to the arrow I8 on the DN scale. The arrow I8 is still in the 0 region of the RN-DZ scale, as it does not reach the +1 region until the dial 'I0 has rotated the distance of one scale division which corresponds to the logarithm of 10 on the slide rule itself.

The hairline is now indicating an answer of .6 on the D scale. The next step of the calculation is to multiply the answer 6 by 3 (the third factor of our example) which is done by moving the left index of the slide under the hairline. This results in no movement of the dial I0 and no movement of the dial 'I4 in relation thereto. It is now desirable to move the rider still farther until the hairline assumes a position over 3 on the C scale, but since this is off scale it is now necessary, in order to arrive at the proper decimal point location, to lock the rider to the stator by pressing on the locking nger 92. This also locks the disk 16. Now by moving the slide I2 back to the left one full scale length, so that the right index thereof is under the hairline, the dial 'I4 is moved counter-clockwise (from the back-Fig. 3) one one full scale division, thus positioning the index I8 in the +1 region, at a position corresponding to the log of 60. In this way an answer falling beyond the right end of the slide rule is taken care of automatically as to Region Number by the dials 'I0 and 14.

It is the dial I4 which rotates counter-clockwise in this instance because of the disk 1.6 being locked, and the rack 2B through the intermediate elements, 34, 32, 36, 38, 44, 46 and 52 rotating the bevel gear 54 clockwise (from the back). The bevel gear 54 itself then rotates the bevel gear I2 and the dial 'I4 through the bevel pinions I8 in the opposite direction, and the same distance.

By releasing the rider by pressure on the unlocking linger and moving the hairline over the 3 on the C scale, the result 18 of the calculation can be read on the D scale. As the rider is moved from 6 on the D scale to 18 on the D scale the dial 'I0 rotates counter-clockwise, thus causing the arrow I8 to move backwards to a position corresponding to the log of 18. In other Words, the arrow I8 is still in the +1 region.

We can now place the decimal point by solving the formula and reading the digits or zeros in the answer on the RN-DZ scale (dial 10) opposite the Difference Number (0) on the DN scale (dial 14). This DZ number is +2 which indicates that there are two digits in the answer, or, in other words, that the answer is 18 and not 1.8 or 180.

In problems in division, the operations are reversed, as the slide rule usually runs oi scale to the left. When the quotient falls beyond the left index of the D scale, the rider is moved to the right index of the C scale and then locked by pressing the nger 92. The slide is then propelled to the right one full scale length from index to index, which results in the dial 'I4 rotating .a full division munten-clockwise (from the back).

It believed that, from the foregoing explanation, the automatic operation oi the dials 'i8 and I4 are obvious for vthose calculations wherein multiplication and division both are present. Whenever the slide is moved until one of its `indices is under the hairline, and the next number in the calculation is beyond the end of the slide rule, it is only necessary to remember to lock the rider against movement by pressure on the finger 92 and move the slide a full scale length in the opposite direction so that said next number in the calculation is within the limits of the D scale. Inmultiplication, the next operation after unlocking the rider is always to move the rider to get the hairline to match .said number so that the 4slide in the rst described rule. i has the RN-DZ scales for first, one-half and oneyrespect to the dial 10a. vates substantially the same as that disclosed in -answer can be read or the calculation continued. lIn division, after the divisor is set under the hairline, the rider is moved to the right index of the C scale and the rider subsequently locked and the slider propelled to the right one full C scale length, V after which the answer can be read on the D scale or the calculation continued.

"v In Figs. 5 and 6 a rotary slide rule is illustrated 'wherein the stator with the D scale thereon is indicated at |09 and the rotor with the C scale thereon'at |221, the rotor corresponding to the The stator Ila third powers printed on the back thereof, and the dial for this scale is indicated as 10a. The DN --scale is printed on a dial 14a which is rotatably mounted by being secured against a shoulder I I4 ofa shaft 98 as by a screw I I2. This shaft has a Vgear |00 mounted thereon, and meshing with a pinion |02 which is freely rotatable on a stud I Gli. The pinion |02 is integral with a gear |06 which meshes with a stationary gear |08, that is, the gear |08 is stationary with relation to the stator Y' l0 by being integral with or secured to a hub I I0 thereof. The stud |04 is carried by a transparent arm IIIa corresponding to the rider in the slide Figs. 3 and 4 for automatically locating the Region Number resulting from calculations on the slide rule without the complication of locking the rider when the answer goes off scale, as in a n straight slide rule. In the rotary type the scale is continuous, and, accordingly, the gearing arrangement for automatically operating the dial 'I0a in relation to the dial 'Hla is very much simplifled. When the rider is rotated a complete revolution the dial l0au has moved a complete scale division and the rider can go on and move an indefinite number of revolutions with the scale divisions adding up or subtracting to determine the Region Number and then the number of digits or zeros, as the case may be, for the answer, by solving the formula This arrangement is limited, of course, to a half revolution of the dial 'IIIEL in either the minus or the plus direction, and the divisions on the DN and RN-DZ scales can be made quite fine if desired for extended computations having a great r number of factors.

From the foregoing specification it is believed obvious how the decimal point locator of Figs. l,

1A, 2 and 2A is operable for manually determining the Region Number and thereby aid in location of the decimal point and how this is automatically done in the forms of invention shown in Figs. 3 and 4 and in Figs. 5 and 6. The factors, digits and zeros in the numerator and the denominator of course must be taken into consideration and calculated in connection with each problem worked out on the slide rule, but this is a relatively simple operation and derives its simplicity from a. formula which has been reduced to the least number of elements. This formula:

v2O is one in which Dn and Dd designate for the answer the total number of digits to the left of the decimal point for those numbers greater than one in the numerator and denominator, respectively, Fn and Fd designate the total number of factors in the numerator and denominator, respectively, and Zu and Zd designate the total number of zeros to the right of the decimal point. This formula omits from consideration that 1 must be subtracted from the formula to get the proper result, but this is taken care of in all decimal point calculations by placing the arrow I8 over l instead of over 0 on the DN scale.

Some changes may be made in the construction and arrangement of the parts of my calculator without departing from the real spirit and purpose of my invention, and it is my intention to cover by my claims any modified forms of struc- Ature or use oi' mechanical equivalents which may be reasonably included within their scope.

I claim as my invention:

1. In a combined slide rule and decimal point locator having the usual logarithmic scales for calculations involving multiplication and division, means associated therewith for indicating region numbers, with all readings within the length of the stator scale of said slide rule considered as falling within one region bearing a designating region number and off scale readings to the right or left of said stator scale considered as falling within other regions having other designating region numbers each representing an additional stator scale length, said means having a pair of scales, one a diiference number scale graduated in units that represent a difference number which is the diierence between the digits minus the factors plus the zeros in the numerator and the digits minus the factors plus the zeros in the denominator of a computation, and the other a region number-digit-zero scale graduated in units representing said region numbers, said pair of scales being movable relative to each other and adapted for the positioning of the difference number scale in relation to the region numberdigit-zero scale, the number oi digits or zeros in the 4answer for said computation being on the region number-digit-Zero scale opposite said the diierence number on the difference number scale.

2. In a combined slide rule and decimal point locator having the usual logarithmic scales, means associated therewith for indicating region numbers, with all readings within the length of the stator scale of said slide rule considered as falling within one region bearing a designating region number and off scale readings to the right or left of said stator scale considered as falling within other regions having other designating region numbers each representing an additional stator scale length, said means having a pair of scales, one a difference number scale graduated in units that represent a dilference number which is the difference between the digits minus the factors plus the zeros in the numerator and the digits minus the factors plus the zeros in the denominator of a computation, and the other a region nu1nber-digitzero scale graduated in units representing said region numbers, said pair of scales being movable relative to each other and adapted for the positioning of the difference number scale in relation to the region numberdigit-zero scale, the number of digits or zeros in the answer for said computation being on the region number-digit-zero scale opposite said the difference number on the difference number scale, said region number-digit-zero scale having readings for both numbers and roots of numbers.

3. A slide rule and decimal point locator having logarithmic scales for performing computations, said slide rule having relatively movable difference number-and region number-digit-Zero scales for cooperation with each other, said difference number scale being graduated in units that represent digits, factors and zeros in the numerator and denominator of a computation according to the formula wherein Dn and Dd designate for the answer the total number of digits to the left of the decimal point for those numbers greater than 1 in the numerator and denominator, respectively, Fn and Fd designate the total number of factors in the numerator and denominator, respectively, and Zn and Zd designate the total number of zeros to the right of the decimal point in those numbers less than 1, said region number-digit-Zero scale being graduated in units representing region numbers with all readings within the length of the stator scale of said slide rule considered as falling within one region bearing a designating region number and off scale readings to the right and left of said stator scale considered as falling within other regions having other designating region numbers each representing an additional stator scale length, and with -1 of the diference number scale opposite the region number on the region number-digit-Zero scale indicating under the deter-mined difference number on the difierence number scale the number of digits or zeros in the answer on the region number-digit-zero scale.

4. A decimal point locator combined with a slide rule comprising means for indicating region numbers corresponding to each scale length off scale reading beyond the end of the slide rule and with all readings within the length of the stator scale considered as falling within one region bearing a designating region number and off scale readings to the right or left of said stator scale considered as falling within other regions bearing other region designating numbers each of which represents an additional stator scale length, and a pair of decimal point deter-- mining scales movable relative to each other, one of said last scales being graduated in units that represent difference numbers and the other being graduated for said region numbers and for digits and zeros, -1 of the difference number scale when opposite the region number on the region number-digit-zero scale, which region number was determined by manipulation of the slide rule, placing opposite the difference number on the difference number scale the number of digits or zeros in the answer on the region number-digit-zero scale said difference number being determined by the formula wherein Dn and Dd designate for the answer the total number of digits to the left of the decimal point for those numbers greater than 1 in the numerator and denominator respectively, Fn and Fd designate the total number of factors in the numerator and denominator respectively, and Zn and Zd designate the total number of zeros to the right of the decimal point in those numbers less than 1.

`5. In a slide rule and decimal point locator, a

stator and a slide having logarithmetic scales, at least one of said scales being repeated a number of times whereby a computation may be performed on the slide rule wherein the answer extends beyond the first scale used to thereby indicate an off scale region, region numbers on said slide rule to indicate both on and off scale regions, a `difference number scale on one element of the slide rule and a region number-digit-zero scale on the other element of the slide rule for cooperation with each other to determine the number of digits or zeros in the answer to the computation by rst determining a diierence number according to the formula wherein Dn and Dd designate for the answer the total number of digits to the left of the decimal point in the numerator and denominator, respectively, Fn and Fd designate the total number of factors in the numerator and denominator, respectively, and Zn and Zd designate the total number of zeros to the right of the decimal point, and then placing -1 of the difference number scale opposite the region number on the region number-digit-zero scale, with all readings within the bounds of said first scale falling within one region bearing a designating region number and o scale readings to the right or left of said iirst scale considered as falling within other regions having other designating region numbers each representing an additional scale length, whereby under the determined difference number on the difference number scale, the number of digits or zeros in the answer is found on the region number-digit-zero scale.

6. In a slide rule and decimal point locator, a stator and a slide having the usual logarithmic scales, at least one of said scales being repeated a number of times and said slide rule having a region number for each of said scales whereby a computation may be performed on said slide rule wherein the answer indicates a region number, a difference number scale on one element of the slide rule and a region number and digit-Zero scale on the other element of the slide rule for cooperation with each other to determine the number of digits or zeros in the answer to said computation by first determining a diierence number according to digits minus factors plus zeros in said computation, whereby under said determined difference number, the number of digits or zeros in the answer may be read on the digitzero scale.

7. In a combined slide rule and decimal point locator, a stator and a slide having repeated logarithmic scales and a rider for calculations involving multiplication and division, means associated therewith for indicating a region number corresponding to said repeated logarithmetic scales, and a pair of scales, one a difference number scale and the other a region number-digit- Zero scale, movable relative to each other and adapted for positioning of -1 of the diiference number scale over the corresponding region number on the region number scale, then determining the number of digits or zeros in the answer for the computation by determining a diierence number depending upon digits minus factors plus zeros in the numerator and denominator of a computation, those of the denominator being subtracted from those of the numerator to determine said difference number, and reading the digits or zeros on the digit-zero scale opposite the determined difference number after said -1 of 23 said diierence number scale has been positioned opposite the determined difference number.

8. In a slide rule having a stator, a slide and a rider, a decimal point locator mounted on said rider and comprisingfa pair of dials, one graduated for Zero, plus and minus diierence numbers and the other for region numbers and digits or zeros in the answer, said dials being geared to said stator and said slide and thereby relatively rotatable during movement of said slide and said rider for determining a region number indicating where successive answers fall both on and beyond -the stator scale of the slide rule, said decimal point locator having means to lock the same for causing relative rotation of said dials when moving said slide whereby the nal region number reading may be taken from the region numberdigit-zero scale at the -1 position on the difierence number scale and opposite the difference number on the difference number scale the digits or zeros in the answer may be read on the region number-digit-Zero scale after said difference number has been determined by the formula wherein Dn and De designate for the answer the total number of digits to the left of the 4decimal point for those numbers greater than 1 in the numerator and denominator respectively, Fn and Fd designate the total number of factors in the numerator and denominator respectively, and Zn and Zd designate the total number of zeros to the right of the decimal point for those numbers less than l.

9. In a slide rule having a stator, a slide and a rider, a decimal point locator mounted on said rider and comprising a pair of dials, one graduated for zero, plus and minus difference numbers and the other for region numbers and digits or zeros in the answer, said dials being geared to said stator and said slide and thereby relatively rotatable during movement of said slide and said rider for determining a region number indicating where successive answers fall both on and beyond the stator scale of the slide rule, said `decimal point locator having means to lock the same for causing relative rotation oi said dials when said stator and said slide are relatively shifted, whereby the nal region number reading may be taken from the region number scale at the -1 position on the difference number scale, and opposite the difference number on the difference number scale the digits or zeros in the answer may be read on the digit-Zero scale after determination of said diierence number depending on digits minus factors plus zeros in the computation.

l0. In a slide rule having a stator, a slide and a rider, a decimal point locator mounted on said rider and comprising a pair of relatively rotatable dials geared to said stator and said slide for indicating a region number for an on-scale region and other region numbers for off-scale regions beyond the ends of the stator scale of the slide rule wherein successive answers in the computation fall either on-scale or oil-scale, said decimal point locator having diierential gearing, and means to lock the same for causing relative rotation of said dials when the computation goes 01T scale and said slide is shifted whereby the final region number reading may be taken from the 'region number scale at a predetermined position on the dilerence number scale and opposite the difference number on the difference number scale the digits or zeros in the answer may be read on the digit-zero scale, said. difference number being determined by the number of digits, minus the number of factors plus the number of zeros or" the numerator and denominator of the computation when those of the denominator are subtracted from those of the numerator.

l1. In a slide rule oi the character disclosed, a stator, a slide, a rider, and a decimal point locator mounted on said rider and comprising a pair .of dials, one having a diierence number scale thereon and the other having a region numberdigit-Zero scale thereon for cooperation with said diierence number scale, the region numbers of said region number-digit-zero scale referring to readings within one region represented by the stator scale and other regions, each a stator scale in length when progressive manipulations of the rule indicate an answer beyond said stator scale and require a reversal of the slide to return the answer to an on-scale position, one of said dials being geared to said stator and the other geared to said slide through the medium of diierential gearing, means for locking said gearing during slide shifts necessary when the reading goes off scale whereby to operate said diierence number and region number-digit-Zero scales in proper relation to each other during calculations, with the final position of one dial relative to the other indicating the region number on the region number scale opposite l on the difference number scale, and opposite the difference number for the computation on the diilerence number scale may be found the number oi digits or zeros in the answer, the difference number being determined by the formula wherein Dn and Dd designate for the answer the total number of digits to the left of the decimal point for those numbers greater than l in the numerator and denominator respectively, Fn and Fd designate the total number of factors in the numerator and denominator respectively, and Zn and Zd designate the total number of zeros to the right of the decimal point.

l2. In a slide rule of the character disclosed, a stator having a rack, a slide having a rack, a rider, and a decimal point locator mounted on said rider and comprising a pair of dials, one of said dials being geared to the rack on said stator and the other geared to the rack on said slide through the medium .of diierential gearing, means for locking said differential gearing against rotation and said rider against movement relative to said stator during slide shifts necessary when the reading goes beyond the stator scale, the nal position of one dial relative to the other indicating regions used in determining the position of the decimal point, one of said regions being the stator scale itseli and others of said regions being one or more stator scale lengths beyond said stator scale.

13. In a rotary slide rule and decimal point locator, a stator and a rotor having the usual logarithmic scales, a rotatable rider arm having a hairline for cooperating with the stator and rotor scales, a pair of dials, one graduated for reading the region numbers and digits or zeros and the other cooperating therewith and graduated for reading dierence numbers, one of said dials being carried by said stator, said stator and the other of said dials being geared together, the gearing including intermediate gears carried by said rider arm whereby to operate said dials in relation to each other by manipulation of the rider arm relative to the stator whereby a region number corresponding to a computation on the slide rule may be read on the region number-digitzero scale opposite -1 on the difference number scale, and the number of digits or zeros in the answer can then be read on the digit-zero scale opposite the difference number, said difference number being determined by the formula wherein Dn and Dd designate for the answer the total number of digits to the left of the decimal point for those numbers greater than 1 in the numerator and denominator respectively, Fn and Fa designate the total number of factors in the numerator and denominator respectively, and Zn and Zd designate the total number of zeros to the right of the decimal point.

14. In a rotary slide rule and decimal point locator, a stator and a rotor having the usual logarithmic scales, a rotatable rider arm, a pair of dials, one graduated for reading region numbers and digits or zeros and the other cooperating therewith and graduated for reading dierence numbers, one of said dials being carried by said stator, said stator and the other lof said dials being geared together, the gearing including intermediate gears carried by said rider arm whereby to operate said dials in relation to each other by rotation of the rider arm whereby a region number corresponding to a computation on the slide rule may be read on the region numberdigit-zero scale in relation to the difference number scale, and the number of digits or zeros in the answer can then be read on the region num- 5;

ber-digit-zero scale opposite the difference number found on the difference number scale, said di'rerence number being determined by the num ber of digits minus the number of factors plus the number of zeros in the computation.

l5. In a rotary slide rule and decimal point locator, a stator and a rotor having the usual logarithmic scales, a rider having a hairline for cooperating with the stator and rotor scales, a pair of dials, one of said dials being carried by said stator, said stator and the other of said dials being geared together, the gearing including ntermediate gears carried by said rider arm whereby to operate said dials in relation to each other by manipulation of the rider relative to the stator whereby a region number corresponding to a computation on the slide rule is indicated on said dials to aid in determining the number of digits or zeros in the answer to a computation on the slide rule, said region number being one of a series of region numbers one of which refers to the stator scale on the rule and the others of which refer to repeated extensions of said scale either to the left `or to the right thereof.

ARTHUR F. ECKEL.

REFERENCES CITED The following references are of record in the le of this patent:

UNITED STATES PATENTS Number Name Date 2,117,413 Gilmore May 17, 1938 2,177,176 Gilmore Oct. 24, 1939 2,328,966 Dickson Sept. 7, 1943 2,363,642 Cherney Nov. 28, 1944 

